The first part of this dissertation deals with the influence of different parameters on the existence of a regular factor in a regular graph. Sharp sufficient conditions for the existence of a regular factor are presented, if either the radius, the chromatic number or the vertex-connectivity of the graph are known, besides the order and the degree of the graph. The second part deals with the question how large the edge-set of a graph can be if the graph has a unique regular factor. The focus will be on extremal bipartite graphs with a unique regular factor. An analysis of the structure shows that all extremal bipartite graphs with a unique k-factor have exactly 2k vertices of minimum degree. This result allows for positive answers on the maximal number of edges in an extremal graph if k is small. This dissertation closes with results on extremal graphs with a unique [1,k]-factor.
info:eu-repo/classification/ddc/510, Mathematik
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